The American Economic Review (Mar. 2008) published research...

The American Economic Review (Mar. 2008) published research on how the gender mix of a U.S. legislator's children can influence the legislator's votes in Congress. Specifically, the researcher investigated how having daughters influences voting on women's issues. The American Association of University Women (AAUW) uses voting records of each member of Congress to compute an AAUW score, where higher scores indicate more favorable voting for women's rights. The researcher modeled AAUW score $(y)$ as a function of the number of daughters $(x)$ a legislator has. Data collected for the 434 members of the 107 th Congress were used to fit the straight-line model, $E(y)=\beta_{0}+\beta_{1} x$. a. If it is true that having daughters influences voting on women's issues, will the sign of $\beta_{1}$ be positive or negative? Explain. b. The following statistics were reported in the article: $\hat{\beta}_{1}=.27$ and $s_{\hat{\beta}_{1}}=.74 .$ Find a $95 \%$ confidence interval for $\beta_{1}$ c. Use the result, part $\mathbf{b}$, to make an inference about the model.

The American Economic Review (Mar. 2008) published research on how the gender mix of a U.S. legislator's children can influence the legislator's votes in Congress. Specifically, the researcher investigated how having daughters influences voting on women's issues. The American Association of University Women (AAUW) uses voting records of each member of Congress to compute an AAUW score, where higher scores indicate more favorable voting for women's rights. The researcher modeled AAUW score $(y)$ as a function of the number of daughters $(x)$ a legislator has. Data collected for the 434 members of the 107 th Congress were used to fit the straight-line model, $E(y)=\beta_{0}+\beta_{1} x$.
a. If it is true that having daughters influences voting on women's issues, will the sign of $\beta_{1}$ be positive or negative? Explain.
b. The following statistics were reported in the article:
$\hat{\beta}_{1}=.27$ and $s_{\hat{\beta}_{1}}=.74 .$ Find a $95 \%$ confidence interval for $\beta_{1}$
c. Use the result, part $\mathbf{b}$, to make an inference about the model.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It estimates parameters such as the intercept and slope coefficients, allowing for predictions and interpretations of how changes in the predictor variable(s) are associated with changes in the outcome variable.

Interpretation of the Slope Coefficient

The slope coefficient in a regression model quantifies the change in the dependent variable for a one-unit change in the independent variable. Its sign (positive or negative) indicates the direction of the relationship: a positive sign means that increases in the independent variable are associated with increases in the dependent variable, while a negative sign indicates an inverse relationship.

Standard Error

The standard error of an estimate measures the variability or dispersion of the estimated parameter across different samples from the same population. It reflects the precision of the estimated coefficient, with a smaller standard error indicating a more precise estimate.

Confidence Interval

A confidence interval provides a range of values within which the true population parameter is likely to lie, based on the sample estimate. It is calculated using the standard error and a critical value from a statistical distribution, thereby quantifying the uncertainty associated with the parameter estimation.

Statistical Inference

Statistical inference involves drawing conclusions about the population based on sample data. In the context of regression analysis, it includes testing hypotheses about the slope coefficient and constructing confidence intervals to assess whether the observed relationship is statistically significant or could have occurred by chance.

Transcript

00:01 For this problem, we are given a lot of background information, but what really matters is that we are told that a researcher modeled a aau score, where a high score indicates more favorable voting for women's rights.

00:17 We're told that, excuse me, we're told that that score is modeled as a function of the number of daughters that a legislator has.

00:26 We're told that data collected for the 434 members of the 107th congress were used to fit the straight line model e of y equals beta not plus beta 1 x.

00:37 In part a, we are asked, if it is true that having daughters influences voting on women's issues, will the sign of beta 1 be positive or negative? and we are asked to explain.

00:48 So the answer is that it would be positive because we would expect that as x in conclusion, so as the number of daughters increases, we would expect that the aau score, or the score for voting positively on women's rights, would go up as well.

01:10 In part b, we are told that the following statistics were reported in the article, that beta 1 equals 0 .27 and s sub -beta 1 equals 0 .74.

01:21 We are asked to find a 95 % confidence interval for beta 1.

01:27 95 % confidence interval will be beta 1 plus or minus t of alpha over 2 in this case this would be t of 0 .025 times s sub beta 1 or so in this case this would be 0 .27 plus or minus now t of 0 .025 is going to be let's see here moment so that would be 0 .27 plus or minus now t of 0 .025 is going to be let's see here moment so that would be 0.

02:01 0 .27 plus or minus 2 .578 times 0 .74...

Please give Ace some feedback